Locally stationary wavelet packet processes : nonstationarity detection and model fitting

For nonstationary time series the fixed Fourier basis is no longer canonical. This article shows how the choice of analysis basis influences the detection of nonstationarities within time series. Rather than limit our basis choice to wavelet or Fourier functions we develop a new stationarity test using a (multiple) bootstrap hypothesis test based on non-decimated wavelet packets. Non-decimated packets are preferred to decimated basis libraries so as to prevent information “loss” at scales coarser than the finest. To complement our stationarity tests we introduce a new class of locally stationary wavelet packet processes and a profile likelihood method to successfully fit these to time series data. We provide new material on the boundedness of the inverse of the inner product oeprator of autocorrelation wavelet packet functions. We demonstrate the effectiveness of our stationarity tests over established methods on real and simulated data.